Punctured Disk (Deleted Disk) & Annulus: Definition

Calculus Definitions >


  1. Punctured Disk
  2. Annulus, How to Find Area

What is a Punctured Disk?

Imagine a flat disk the size of the milky way, then take a (very thin) pin and prick through the disk in the exact center; You’ve just created a punctured disk (sometimes called a deleted disk).

punctured disk 2
Punctured open disk around z0(left); Open disk around z0 (right).

More technically, this type of disk is an annulus (a flat donut) with an inner radius of zero, i.e. a disk without a center. However, there is more than one definition. For example Nevanlinna (1970) regarded the point of puncture as when a boundary curve shrinks to a single point. This makes sense when you think of a black hole, which is a gravitational singularity with an enormous mass in an infinitely small space. Disks can also be punctured more than once (Beardon, 1983), so you may see them described as once-punctured or twice punctured.

Punctured disks arise in complex analysis when studying functions that are well-behaved except for a central point, where there may be an isolated singularity.

Describing the Punctured Disk

A punctured disk can be described as an open annulus with center x0 and an inner radius of zero. The region is defined by two inequalities

0 < |z – z0| < R,

where R = the punctured disk’s radius.

The radius can be any positive number; when R is infinity, the region is usually called a punctured plane (Sarason, 2007).

A few interesting features of punctured disks:

  • Although punctured disks are sometimes described as an open annulus, the disks themselves are neither closed sets nor open sets,
  • The disks aren’t simply connected. The puncture prevents a closed curve around the region from contracting to a point while keeping within the region (MIT).
  • A punctured disk can also be written in terms of a Laurent series.

Annulus Definition

The annulus is the region between two concentric rings—two circles that share the same center. This flat donut shape, or washer, is important for many engineering applications.

The center of mass for an annulus is not within the shape, but at the shared center of both circles. In the image below, the center of mass is labeled as “C”.
center of mass annulus

Area of an Annulus

To calculate the area, think about the shape as a larger circle with a smaller circle taken out of it. If the radius of the larger circle is R and the radius of the smaller circle is r, the radius of the circle will be π(R2 – r2). In the following image, the large ring has a radius of 4 and the smaller ring has a radius of 2:

You can get to this same result using calculus if you imagine dividing up your annulus into many many annuli (very thin donuts) with an infinitesimal width. The formula is:

Here, you integrate over very thin rings of width dρ, where ρ is the radius of the inner ring. Notice the result is the same as that which you’d reach by using the simple geometrical method above.

Since the full circle is 360 degrees, the area of a M degree piece of an annulus would be given by:
π(R2 – r2)(M/360).

Finding the Area of an Annulus: Example

Example question: Find the area of a 90 degree slice (a quarter) of an annulus where the radius of the outer ring is 2 and the radius of the inner circle is 1.

Step 1: Identify the parts you need for the formula from the question:

  • M (degrees) = 90 degrees,
  • R (radius of outer ring) = 2,
  • r (radius of inner ring) = 1.

Step 2: Plugg those numbers into the formula:

  • π(R2 – r2)(M/360)
  • = π (22 – 12)(90/360), or  π(3)(90/360)
  • = 3π/4.

Annulus in Complex Analysis

In complex analysis, you can describe an annulus by writing ann(a; r, R), where:

  • a is the center,
  • r is the inner radius,
  • R is the outer radius.

It is defined as an open region such that

r < |z-a| < R

An annulus with an inner radius of zero is called a punctured disk; it is a filled circle with the (infinitesimally small) central point missing.

Punctured Disk: References

Beardon, A. (1983). The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer-Verlag, Berlin.
MIT OpenCourseware. (2010). Retrieved September 27, 2020 from: https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/3.-double-integrals-and-line-integrals-in-the-plane/part-c-greens-theorem/session-72-simply-connected-regions-and-conservative-fields/MIT18_02SC_pb_72_comb.pdf
Nevanlinna, R. (1970). Analytic functions, Springer-Verlag, Berlin.
Singularities. Article Posted on website The Physics of The Universe.
Page, John. Calculating the Area of Annulus. Retrieved from http://www.mathopenref.com/annulusarea.html on September 28, 2018.
Sarason, D. (2007). Complex Function Theory. American Mathematical Society.

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